Research Article
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A Note on Equivalence of G-Cone Metric Spaces and G-Metric Spaces

Year 2023, Issue: 43, 73 - 82, 30.06.2023

Abhishikta Das Tarapada Bag

https://doi.org/10.53570/jnt.1277026

Abstract

This paper contains the equivalence between tvs-G cone metric and G-metric using a scalarization function $\zeta_p$, defined over a locally convex Hausdorff topological vector space. This function ensures that most studies on the existence and uniqueness of fixed-point theorems on G-metric space and tvs-G cone metric spaces are equivalent. We prove the equivalence between the vector-valued version and scalar-valued version of the fixed-point theorems of those spaces. Moreover, we present that if a real Banach space is considered instead of a locally convex Hausdorff space, then the theorems of this article extend some results of G-cone metric spaces and ensure the correspondence between any G-cone metric space and the G-metric space.

Keywords

G-metric space G-cone metric space tvs-G cone metric space

References

  • S Gahler, \emph{2-Metrische Raume Und Ihre Topologische Struktur}, Mathematische Nachrichten 26 (1-4) (1963) 115--148.
  • S. Czerwik, \emph{Contraction Mappings in b-Metric Spaces}, Acta Mathematica et Informatica Universitatis Ostraviensis 1 (1) (1993) 5--11.
  • S. Czerwik, \emph{Nonlinear Set-Valued Contraction Mappings in b-Metric Spaces}, Atti del Seminario Matematico e Fisico dell'Universita di Modena 46 (1998) 263--276.
  • W. Kirk, N. Shahzad, Fixed Point Theory in Distance Spaces, Springer, Cham, 2014.
  • B. C. Dhage, \emph{Generalized Metric Spaces Mappings with Fixed Point}, Bulletin of Calcutta Mathematical Society 84 (1992) 329--336.
  • Z. Mustafa, B. Sims, \emph{A New Approach to Generalized Metric Spaces}, Journal of Nonlinear Convex Analysis 7 (2) (2006) 289--297.
  • S. Sedghi, N. Shobe, A. Aliouche, \emph{A Generalization of Fixed Point Theorems in S-Metric Spaces}, Matematoqki Vesnik 64 (3) (2012) 258--266.
  • H. L. Guang, Z. Xian, \emph{Cone Metric Space and Fixed Point Theorems of Contractive Mapping}, Journal of Mathematical Analysis and Applications 322 (2) (2007) 1468--1476.
  • A. Das, T. Bag, \emph{A Study on Parametric S-metric Spaces}, Communications in Mathematics and Applications 13 (3) (2022) 921--933.
  • A. Das, T. Bag, \emph{A Generalization to Parametric Metric Spaces}, International Journal of Nonlinear Analysis and Application 14 (1) (2023), 229–244.
  • G. Rano, T. Bag, S.K. Samanta, \emph{Fuzzy Metric Space and Generating Space of Quasi-Metric Family}, Annals of Fuzzy Mathematics and Informatics 11 (2016) 183--195.
  • I. Beg, M. Abbas, T. Nazir, \emph{Generalized Cone Metric Spaces}, The Journal of Nonlinear Sciences and Its Applications 3 (1) (2010) 21--31.
  • G. Y. Chen, X. X. Huang, X. Q. Yang, Vector Optimization: Set-Valued and Variational Analysis, Springer Berlin, Heidelberg, 2005.
  • W. S. Du, \emph{On Some Nonlinear Problems Induced by an Abstract Maximal Element Principle}, Journal of Mathematical Analysis and Applications 347 (2) (2008) 391--399.
  • C. Gerth, P. Weidner, \emph{Nonconvex Separation Theorems and Some Applications in Vector Optimization}, Journal of Optimization Theory and Applications 67 (2) (1990) 297--320.
  • A. Göpfert, C. Tammer, C. Z. Linescu, \emph{On the Vectorial Ekeland's Variational Principle and Minimal Points in Product Spaces}, Nonlinear Analysis: Theory, Methods and Applications 39 (7) (2000) 909--922.
  • A. Göpfert, C. Tammer, H. Riahi, C. Z. Linescu, Variational Methods in Partially Ordered Spaces, Springer, New York, 2003.
  • W. Du, \emph{A Note on Cone Metric Fixed Point Theory and Its Equivalence}, Nonlinear Analysis: Theory, Methods and Applications 72 (5) (2010) 2259--2261.
  • I. J. Lin, C. M. Chen, C. H. Chen, T. Y. Cheng, \emph{A Note on tvs-G-Cone Metric Fixed Point Theory}, Journal of Applied Mathematics 2012 (2012) Article ID 407071 10 pages.
  • Z. Mustafa, H. Obiedat, F. Awawdeh, \emph{Some Fixed Point Theorem for Mapping on Complete G-Metric Spaces}, Fixed Point Theory and Applications 2008 (2008) Article Number 189870 12 pages.

Year 2023, Issue: 43, 73 - 82, 30.06.2023

Abhishikta Das Tarapada Bag

https://doi.org/10.53570/jnt.1277026

Abstract

References

  • S Gahler, \emph{2-Metrische Raume Und Ihre Topologische Struktur}, Mathematische Nachrichten 26 (1-4) (1963) 115--148.
  • S. Czerwik, \emph{Contraction Mappings in b-Metric Spaces}, Acta Mathematica et Informatica Universitatis Ostraviensis 1 (1) (1993) 5--11.
  • S. Czerwik, \emph{Nonlinear Set-Valued Contraction Mappings in b-Metric Spaces}, Atti del Seminario Matematico e Fisico dell'Universita di Modena 46 (1998) 263--276.
  • W. Kirk, N. Shahzad, Fixed Point Theory in Distance Spaces, Springer, Cham, 2014.
  • B. C. Dhage, \emph{Generalized Metric Spaces Mappings with Fixed Point}, Bulletin of Calcutta Mathematical Society 84 (1992) 329--336.
  • Z. Mustafa, B. Sims, \emph{A New Approach to Generalized Metric Spaces}, Journal of Nonlinear Convex Analysis 7 (2) (2006) 289--297.
  • S. Sedghi, N. Shobe, A. Aliouche, \emph{A Generalization of Fixed Point Theorems in S-Metric Spaces}, Matematoqki Vesnik 64 (3) (2012) 258--266.
  • H. L. Guang, Z. Xian, \emph{Cone Metric Space and Fixed Point Theorems of Contractive Mapping}, Journal of Mathematical Analysis and Applications 322 (2) (2007) 1468--1476.
  • A. Das, T. Bag, \emph{A Study on Parametric S-metric Spaces}, Communications in Mathematics and Applications 13 (3) (2022) 921--933.
  • A. Das, T. Bag, \emph{A Generalization to Parametric Metric Spaces}, International Journal of Nonlinear Analysis and Application 14 (1) (2023), 229–244.
  • G. Rano, T. Bag, S.K. Samanta, \emph{Fuzzy Metric Space and Generating Space of Quasi-Metric Family}, Annals of Fuzzy Mathematics and Informatics 11 (2016) 183--195.
  • I. Beg, M. Abbas, T. Nazir, \emph{Generalized Cone Metric Spaces}, The Journal of Nonlinear Sciences and Its Applications 3 (1) (2010) 21--31.
  • G. Y. Chen, X. X. Huang, X. Q. Yang, Vector Optimization: Set-Valued and Variational Analysis, Springer Berlin, Heidelberg, 2005.
  • W. S. Du, \emph{On Some Nonlinear Problems Induced by an Abstract Maximal Element Principle}, Journal of Mathematical Analysis and Applications 347 (2) (2008) 391--399.
  • C. Gerth, P. Weidner, \emph{Nonconvex Separation Theorems and Some Applications in Vector Optimization}, Journal of Optimization Theory and Applications 67 (2) (1990) 297--320.
  • A. Göpfert, C. Tammer, C. Z. Linescu, \emph{On the Vectorial Ekeland's Variational Principle and Minimal Points in Product Spaces}, Nonlinear Analysis: Theory, Methods and Applications 39 (7) (2000) 909--922.
  • A. Göpfert, C. Tammer, H. Riahi, C. Z. Linescu, Variational Methods in Partially Ordered Spaces, Springer, New York, 2003.
  • W. Du, \emph{A Note on Cone Metric Fixed Point Theory and Its Equivalence}, Nonlinear Analysis: Theory, Methods and Applications 72 (5) (2010) 2259--2261.
  • I. J. Lin, C. M. Chen, C. H. Chen, T. Y. Cheng, \emph{A Note on tvs-G-Cone Metric Fixed Point Theory}, Journal of Applied Mathematics 2012 (2012) Article ID 407071 10 pages.
  • Z. Mustafa, H. Obiedat, F. Awawdeh, \emph{Some Fixed Point Theorem for Mapping on Complete G-Metric Spaces}, Fixed Point Theory and Applications 2008 (2008) Article Number 189870 12 pages.
There are 20 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Article
Authors

Abhishikta Das 0000-0002-2860-424X

Tarapada Bag 0000-0002-8834-7097

Publication Date June 30, 2023
Submission Date April 4, 2023
Published in Issue Year 2023 Issue: 43

Cite

APA Das, A., & Bag, T. (2023). A Note on Equivalence of G-Cone Metric Spaces and G-Metric Spaces. Journal of New Theory(43), 73-82. https://doi.org/10.53570/jnt.1277026
AMA Das A, Bag T. A Note on Equivalence of G-Cone Metric Spaces and G-Metric Spaces. JNT. June 2023;(43):73-82. doi:10.53570/jnt.1277026
Chicago Das, Abhishikta, and Tarapada Bag. “A Note on Equivalence of G-Cone Metric Spaces and G-Metric Spaces”. Journal of New Theory, no. 43 (June 2023): 73-82. https://doi.org/10.53570/jnt.1277026.
EndNote Das A, Bag T (June 1, 2023) A Note on Equivalence of G-Cone Metric Spaces and G-Metric Spaces. Journal of New Theory 43 73–82.
IEEE A. Das and T. Bag, “A Note on Equivalence of G-Cone Metric Spaces and G-Metric Spaces”, JNT, no. 43, pp. 73–82, June 2023, doi: 10.53570/jnt.1277026.
ISNAD Das, Abhishikta - Bag, Tarapada. “A Note on Equivalence of G-Cone Metric Spaces and G-Metric Spaces”. Journal of New Theory 43 (June 2023), 73-82. https://doi.org/10.53570/jnt.1277026.
JAMA Das A, Bag T. A Note on Equivalence of G-Cone Metric Spaces and G-Metric Spaces. JNT. 2023;:73–82.
MLA Das, Abhishikta and Tarapada Bag. “A Note on Equivalence of G-Cone Metric Spaces and G-Metric Spaces”. Journal of New Theory, no. 43, 2023, pp. 73-82, doi:10.53570/jnt.1277026.
Vancouver Das A, Bag T. A Note on Equivalence of G-Cone Metric Spaces and G-Metric Spaces. JNT. 2023(43):73-82.
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